Let \(Q=(q_0,\dots ,q_n)\)  be an \((n+1)\) -tuple of positive integers such that \(p\nmid q_i\) , \(0\le i\le n\) , and \(\gcd (q_0,\dots , \hat{q_i},\dots ,q_n)=1\)  for every \(0\le i \le n\) , where \(\hat{q_i}\)  means that \(q_i\)  is omitted. The weighted projective \(n\) -space over \(k\)  of type \(Q\) , denoted by \(\mathbb {P}^n(Q)\) , is the projective variety \(\mathbb {P}^n(Q):=\operatorname{Proj} k[x_0,\dots ,x_n]\) , where the polynomial algebra is graded by the condition \(\deg (x_i)=q_i\)  for \(0\le i\le n\)  (cf. [1]}).
